Integrand size = 29, antiderivative size = 80 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {(A b-a B) x}{a^2+b^2}+\frac {A \log (\sin (c+d x))}{a d}-\frac {b (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a \left (a^2+b^2\right ) d} \]
-(A*b-B*a)*x/(a^2+b^2)+A*ln(sin(d*x+c))/a/d-b*(A*b-B*a)*ln(a*cos(d*x+c)+b* sin(d*x+c))/a/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {\frac {(A+i B) \log (i-\tan (c+d x))}{a+i b}-\frac {2 A \log (\tan (c+d x))}{a}+\frac {(A-i B) \log (i+\tan (c+d x))}{a-i b}+\frac {2 b (A b-a B) \log (a+b \tan (c+d x))}{a \left (a^2+b^2\right )}}{2 d} \]
-1/2*(((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) - (2*A*Log[Tan[c + d*x]] )/a + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)))/d
Time = 0.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3042, 4094, 3042, 25, 3956, 4013}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x) (a+b \tan (c+d x))}dx\) |
\(\Big \downarrow \) 4094 |
\(\displaystyle -\frac {b (A b-a B) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {A \int \cot (c+d x)dx}{a}-\frac {x (A b-a B)}{a^2+b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b (A b-a B) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {A \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {x (A b-a B)}{a^2+b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {b (A b-a B) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {A \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {x (A b-a B)}{a^2+b^2}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {b (A b-a B) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {x (A b-a B)}{a^2+b^2}+\frac {A \log (-\sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 4013 |
\(\displaystyle -\frac {b (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac {x (A b-a B)}{a^2+b^2}+\frac {A \log (-\sin (c+d x))}{a d}\) |
-(((A*b - a*B)*x)/(a^2 + b^2)) + (A*Log[-Sin[c + d*x]])/(a*d) - (b*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d)
3.3.71.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*tan[(e_.) + (f_. )*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(B*(b* c + a*d) + A*(a*c - b*d))*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[b*((A*b - a*B)/((b*c - a*d)*(a^2 + b^2))) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] + Simp[d*((B*c - A*d)/((b*c - a*d)*(c^2 + d^2))) Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19
method | result | size |
parallelrisch | \(\frac {\left (-2 A \,b^{2}+2 B a b \right ) \ln \left (a +b \tan \left (d x +c \right )\right )+\left (-A \,a^{2}-B a b \right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+2 A \left (a^{2}+b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-2 a d x \left (A b -B a \right )}{2 \left (a^{2}+b^{2}\right ) a d}\) | \(95\) |
derivativedivides | \(\frac {\frac {\frac {\left (-a A -B b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A b +B a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a}-\frac {\left (A b -B a \right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a}}{d}\) | \(101\) |
default | \(\frac {\frac {\frac {\left (-a A -B b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A b +B a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a}-\frac {\left (A b -B a \right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a}}{d}\) | \(101\) |
norman | \(-\frac {\left (A b -B a \right ) x}{a^{2}+b^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a d}-\frac {\left (a A +B b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {\left (A b -B a \right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a d}\) | \(106\) |
risch | \(-\frac {x B}{i b -a}-\frac {i x A}{i b -a}-\frac {2 i x A}{a}-\frac {2 i A c}{a d}+\frac {2 i b^{2} A x}{a \left (a^{2}+b^{2}\right )}+\frac {2 i b^{2} A c}{a d \left (a^{2}+b^{2}\right )}-\frac {2 i b B x}{a^{2}+b^{2}}-\frac {2 i b B c}{d \left (a^{2}+b^{2}\right )}+\frac {A \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{a d \left (a^{2}+b^{2}\right )}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{d \left (a^{2}+b^{2}\right )}\) | \(240\) |
1/2*((-2*A*b^2+2*B*a*b)*ln(a+b*tan(d*x+c))+(-A*a^2-B*a*b)*ln(sec(d*x+c)^2) +2*A*(a^2+b^2)*ln(tan(d*x+c))-2*a*d*x*(A*b-B*a))/(a^2+b^2)/a/d
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {2 \, {\left (B a^{2} - A a b\right )} d x + {\left (A a^{2} + A b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (B a b - A b^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \]
1/2*(2*(B*a^2 - A*a*b)*d*x + (A*a^2 + A*b^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + (B*a*b - A*b^2)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)))/((a^3 + a*b^2)*d)
Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 952, normalized size of antiderivative = 11.90 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
Piecewise((zoo*x*(A + B*tan(c))*cot(c)/tan(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-A*log(tan(c + d*x)**2 + 1)/(2*d) + A*log(tan(c + d*x))/d + B*x)/a , Eq(b, 0)), ((-A*x - A/(d*tan(c + d*x)) - B*log(tan(c + d*x)**2 + 1)/(2*d ) + B*log(tan(c + d*x))/d)/b, Eq(a, 0)), (A*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) - I*A*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) - I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) - A*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) - 2*I*b*d) + 2*I*A*log(tan(c + d*x))*ta n(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + 2*A*log(tan(c + d*x))/(2*b*d*t an(c + d*x) - 2*I*b*d) + A/(2*b*d*tan(c + d*x) - 2*I*b*d) + I*B*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + B*d*x/(2*b*d*tan(c + d*x) - 2*I*b* d) + I*B/(2*b*d*tan(c + d*x) - 2*I*b*d), Eq(a, -I*b)), (A*d*x*tan(c + d*x) /(2*b*d*tan(c + d*x) + 2*I*b*d) + I*A*d*x/(2*b*d*tan(c + d*x) + 2*I*b*d) + I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) - A*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) + 2*I*b*d) - 2*I*A*log(ta n(c + d*x))*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) + 2*A*log(tan(c + d*x))/(2*b*d*tan(c + d*x) + 2*I*b*d) + A/(2*b*d*tan(c + d*x) + 2*I*b*d) - I*B*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) + B*d*x/(2*b*d*tan(c + d*x) + 2*I*b*d) - I*B/(2*b*d*tan(c + d*x) + 2*I*b*d), Eq(a, I*b)), (x*(A + B*tan(c))*cot(c)/(a + b*tan(c)), Eq(d, 0)), (-A*a**2*log(tan(c + d*x)**2 + 1)/(2*a**3*d + 2*a*b**2*d) + 2*A*a**2*log(tan(c + d*x))/(2*a**3*d + ...
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.34 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\frac {2 \, {\left (B a - A b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {2 \, {\left (B a b - A b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3} + a b^{2}} - \frac {{\left (A a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, A \log \left (\tan \left (d x + c\right )\right )}{a}}{2 \, d} \]
1/2*(2*(B*a - A*b)*(d*x + c)/(a^2 + b^2) + 2*(B*a*b - A*b^2)*log(b*tan(d*x + c) + a)/(a^3 + a*b^2) - (A*a + B*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) + 2*A*log(tan(d*x + c))/a)/d
Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\frac {2 \, {\left (B a - A b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {{\left (A a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3} b + a b^{3}} + \frac {2 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a}}{2 \, d} \]
1/2*(2*(B*a - A*b)*(d*x + c)/(a^2 + b^2) - (A*a + B*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) + 2*(B*a*b^2 - A*b^3)*log(abs(b*tan(d*x + c) + a))/(a^3*b + a*b^3) + 2*A*log(abs(tan(d*x + c)))/a)/d
Time = 9.40 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.44 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,b-B\,a\right )}{a\,d\,\left (a^2+b^2\right )} \]